889 equitable assignment rules(u) wisconsin univ …c key words: assignment rules, dynamic...
TRANSCRIPT
D-R142 889 EQUITABLE ASSIGNMENT RULES(U) WISCONSIN UNIV-IIADISON i/iMATHEMATICS RESEARCH CENTER P GLYNN ET AL. MAY 84NRC-TSR-2685 DAAG29-88 C-0841
UNCLASS7FIED F/G 12/ NL
EEEmEEmhEEmhEILflflflflflflflfl.. "
It
.=m . .. ,, z ,: *u- .. . , :: . -. . . * .i '* . ... *-o* - .' - - .. o '-. , 1. . ' '
ILI &3L-2'
.. .D0
JL6'
MICROCOPY RESOLUTION TEST CHART q---
NI BR ,OF S
*.-' .r
'miltil* 5 .i
- ' "- -- -e
P'. Gln an .L. Sander
Mathematics Research CenterUniversity of Wisconsin-Madison610 Walnut StreetMadison, Wisconsin 53705
May 1984
(Received March 1, 1984)
LAJApproved for public release
La-. Distribution unlimited
* Sponsored by
U. S. Army Research OfficeJU1o 84P. 0. Box 12211Research Triangle ParkNorth Carolina 27709
8 4 G
.UNIVERSITY OF WISCONSIN-MADISONMATHEMATICS RESEARCH CENTER
EQUITABLE ASSIGNMENT RULES
P. Glynn and J. L. Sanders
Technical Summary Report #2685May 1984
ABSTRACT
This paper investigates the formalization of an important class of
management decision problems. The problems considered are those of making
equitable workload assignments to personnel. The paper proposes a series of
intuitively appealing assignment rules, including random assignment, fixed
assignment, block rotation and rules that reverse inequities caused by the
last period's assignments. It is shown that in the two-person case none of
these rules satisfies the simple criterion that cumulative differences of
workload assignments among personnel become and remain small. Differences in
the properties of these rul.-s are investigated under three additional but less
strenuous criteria. It is shown that a new assiqnment rule called the
"counter-current" rule does satisfy the criterion stated above; further, it is
shown that it is an optimal rule under a fairly weak set of requirements. The
extension of the results from the two-person case to the n-person case to the
n-person case is discussed briefly and some initial results are presented.
A14S (MOB) Subject Classifications: 49C20, 60K10, 93E20
C Key Words: Assignment rules, dynamic programming
Work Unit Number 5 (Optimization and Large Scale Systems)
Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.
_--,--
, k,%, ., , , ,%.,
'"- " "' * - ' ", '.'' , >*t'. .'" "-. °- . . . . "- *- *.'' ' " " '- - , - . - ', " ",,,. .% • % • ,. o, .. % .. . . . .. oO .. . ... .- . • . . . . . . . .- - . ,.
.,., a
•.•
SIGNIFICANCE AND EXPLANATION
Consider the following assigment problem. A car rental company wishes
to assign cars in such a way so as to eoualize wear. The assignment rule must
reflect the fact that the wear introduced by a customer is a random
quantity. A similar situation is faced by a manager who must assign random
workloads to employees in an equitable manner. In this paper, we consider
decision rules for problems of the above type. We show that while a number of
commonly used decision rules have undesirable asymptotic properties, there
exists a rule, which we call countercurrent, that has good long-run
characteristics and is optimal for a reasonable criterion function.
Furthermore, the countercurrent decision rule is easy to implement.
$ 1 0
The responsibility for the wording and views expressed in this descriptivesuimary lies with NRC, and not with the authors of this report.
I I.. .. ...... .. .. . ... .. . ,. .IN, . .o, .. .oo . ,. . . o . . O. o .-. ~ ,.o . " .
EQUITABLE ASSIGNMENT RULES
P. Glynn and J. L. Sanders
1. Introduction
Consider a facility with two employees that is required to process two tasks
per day. Assume that each task has an associated index of effort and that
the task effort indices form stochastic sequences. The goal of the facility
manager is to assign tasks to employees in as "equitable" a manner as
possible.
The concept of equitable treatment of employees is common in the organiza-
tional behavior literature. In particular, the perceived equity of pay
received for work delivered has received attention. According to Shapiro and
Wahba (1978), "Dissatisfaction (with pay) results in many dysfunctional
reactions such as turnovers, absenteeism, alcohol and drug problems, union
formations, strikes, slowdowns, decreased performance, grievances, increased
training costs, high accident rates and in turn increased unemployment and
accident insurance costs." Also of interest are definitions of perceived
equity based on perceptions of the ratio of rewards to the individual from
the organization to the input effort provided by the individual and the com-
parison of these reward ratios among individuals in the organization. (Adams
(1963 and 1965))
Shapiro and Wshba (1978) in their empirical study, show that social comparison
of pay received for work done is the single most important variable in
explaining pay dissatisfaction. Andrews and Henry (1963), who surveyed
managers who ware dissatisfied with their pay, report that 87% of those
managers felt that their subordinates had a better outcome/input ratio than
Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.
- .i they did. Lalr (1971) reports similar results. In the rea of ay sts-
:' .ifaetion. perceived equity lealy Is an importnt. nd possibly the preier
4.'
Ceuny ations with nuber o directors o nursing in acute are hospitals
indicate that perceived inequities of work assignments among staff nurses can
profoundly disrupt a nursing unit. resulting in absenteeism, turnover and
other problems cited by Shapiro and ahba. In this case, the equity issue
surfaoes around work assignment alone. aside from any differences in pay.
In a related setting, Adams (1963) has developed a theory of motivation
related to perceived equity of rewards. He explores the implications of
equity for prediction of individual behavior in business organizations.
In our case, we assume that the manager is only free to adjust work assign-
ments. Thus, the equity issue refers to equalization of the work effort
assigned to the employee pool.
(1.1) Example. A director of nursing manages two nurses and two nursing
units. Asume that the difficulty of each assignment is given by the combined
patient care acuity measure associated with the patients on a unit for the
shift of duty being considered.
iursing care acuity measures are systems that survey the patient's need for
clinical care, assistance with daily living (e.g. feeding, bathing, etc.),
education, emotional support and other aspects of nursing care. They are not
designed to assess the severity of Illness of the patient. They are Instead
used primarily to assess the mount and level of nursing care required.
-2-
% I• r V e C..-
- .5, -4 -. p-, . .-- : . * * .- 07 1 V. T 7 T ,. - . 7. -
Clearly, as patients come and go. the sum of the acuity levels of patients on
a unit may fluctuate unpredictably. In some cases, the acuity assessments
made on a unit at the beginning of a shift will remain accurate over the
entire shift. In this case we know in advance the level of effort and skill
required to staff that unit on that shift. In other cases, such as psychia-
trio and geriatio units, the assessments made at the beginning of the shift
may be grossly inaccurate at the end of the tour. Older patients may become
confused and psychiatric patients may experience an exacerbation of their con-
dition, thus requiring higher levels of attention. In the first Case. we
assume that we know the acuity requirement in advance of the work assignment
event though that level varies stochastically from assignment period to
assignment period. In the second Case. we cannot predict the level of acuity
associated with an assignment in advance, although we may know the average
value of acuity associated with the assignment.
The problem of equalizing effort also arises outside of the nursing context
desoribed above.
(1.2) Example. The manager of a typing pool receives two typing assign-
meats per day. The goal of the manager is to assign the typing tasks among
the two available typists so as to equalize the mount of work assigned to the
employees over a given time horizon.
(1.3) Example. Consider a factory production unit with two machines. On
each day, the unit is required to process two tasks. The goal of the produc-
tion manager is to assign tasks to the available machines so as to equalize
-3-
.. -1 -
(1.4) Example. The manager of a car rental company Wishes to assign the
oampany's two cars to customers so as to equalize mileage.
In this paper, we will analyze decision rules for the above task assignment
problems. Section 2 develops the mathematical framework for the problem and
discusses basic properties of several common, intuitively appealing decision
rules. In Section 3. we examine a rule, which we call counter-current, which
is optimal with respect to a number of different criteria. Section 4 is
devoted to the finer stochastic properties of the counter-current decision
rule. Finally, in Section 5, we briefly discuss the n-task, n-person case.
and offer some concluding remarks.
2. The Class of Task Assignment Rules
In order to develop a mathematical framework for the task assignment problem,
we consider Exmple 1.1. Let the acuity measures on day n for the two
nursing units be given by Vn and Wn . We assume, throughout this paper, that:
l £1, a(,nW Vn):n > 1) is a sequence of independent, identically
distributed (L.i.d.) random vectors (r.v.'s)
A2. there exists K < -such that P(ID n <K : 1
£3. P(D < x} has a density which is positive on [-K,K]
where D V -Wa n n
A4. EDa > 0.
-4-
4-4
f 2€ .' .'. ."". "" ."" '* " ' ' * " * a" " % " -" ". ."""4"" . ... , . . .. •".. .. ... . . . , , , -
We also suppose that there exists an independent sequence (U :n > 1) of i.i.d.n
uniform r.v.'s.
A decision rule is a sequence (aa:n !. 1) of r.v.'s taking values in (0,1).
A decision rule is said to be a randomized non-anticipating decision rule
it
C (U ap~ 1 1( n- n-I n-i Ct fVW ,. V ,W L V , W% ~ ~ ~ ~ ~ a annl -. al
for some Borel-measurable function f. and real number p, (IA denotes a r.v.
which is 1 or 0 depending on whether or not A has occurred). A rule is said
to be randomized and strictly non-anticipating if it can be represent-
ed as
a I{ < Pa ) f (Vt. W1 .... V -l' W , Cn-1
Set
A . a ir_ + (U - a ilW i
a
Sn "E (I - aI)V, + a iW.
If w follow the nursing unit example, we interpret X and Y as the cumula-
tive imount of task effort assigned to nurses I and 2, respectively, by time
•. % .. .. . ". " ". " -. .. . - .. - .. % ,,, -% " .,, . . % t% ""% . % ,• . ... . .. ." , . .
% % %q e . * ~ * * **'. *B % .
(2.1) Example- Set M 1. Since ED a> 0, Z n a.3. wherezua a u -
Such a decision rule is clearly inequitable, due to an obvious lack of
symmetry. Any "good" rule should be symmetric in the sense that for all n.
Pfu ORn, W) n 1/2 a.
Itr Vu(*....,V ), W (W ....,W ).Other desirable properties at a
decision rule are:
(P1). the r.v. Iz nI should grow as slowly as possible
0P2). ZV! I X I ) should be as sall as Possible, where
% - nfa> :. <.4)
(P) 316/ should be as close to 1/2 as possible, where
n
Defore proceeding to a discussion of optimal decision rules in Section 3, we
first exine five decision rules that either have been used in practice for
scheduling or are intuitively appealing as asaigment rules.
-6-
A, *.C e-e
+._, .
(1). n IA whereA- (U C1/2)
02. Lan Ozn~ l ' A' -i where A- (U, < 1/2)
(13). l - "A where An -(U < 1/2 )' a
(M4:). a - 13, where B - (U < 1/2). - (Z < Z 2 }, for a > 2.n - - n n11 - -2)
(R51. 1n" B w here 5 (U < 1/2), Bu (zn <. >} 0)a I - n -1 - az2' n
(z z, D < 0) for , > 2.
,J
Rule R1 Is equivalent to making the first assignment of personnel to duties
~-7-
• " "'Q °%e e" " w .,. " - " * ..". ."- • • . .-rdi ''. " '. W, ql e . " , . - % . . + ".". •...+.•...+..,t+ . .,-. ;,. ,,'"" " " '" ,"; "" " '' "'" . """ ' '' ";" '' " '' """ "-""-V N .;'''"-"".;. '.%+
*10
by flipping a fair coin. However, once these assignments are made they are
assamed to remain there indefinitely. In other words, if nurse "A" draws Unit
2 on the first assignment it is assumed that she/he remains on assignment
there in the future. This rule is "fair" in the sense that both nurses have
* equal probability of being assigned to a particular unit.
Rule R2 makes the first assignment as in Rule R1 but thereafter strictly
rotates the nurses between assignments. This rule is a trivial example of
the "Block Rotation" systems that have been used in personnel scheduling for
sme time. The rule is "fair" in the sense that both nurses are assigned to a
given duty station the same fraction of the time.
Rule 23 assigns personnel to duties by extending the randomization employed
at the beginning of Rule R1 to every period. This "purely random" rule could.
be realized In practice by repeated application of the "coin tossing"
mechanisn and may be approximated by haphazard processes where the assignment
*. mechanis, employed no previous memory of previous assignments or their
outcomes.
Rules R4 and R5 look to the previous period and examine the outcome of the
assignments of that period. The assignment made in this period attempts to
reverse the inequities of the previous period. In R5. the rule is "clair-
voYant*; it assumes knowledge of the true acuity measures for the shift before
the assignments are made. Rule R4 is the non-clairvoyant version of R5, in
Wiloh the outcome of the assignment is unpredictable. Both are "fair"--they
try to continually reverse any inequities that arose from the previous
%
6 %
4,.
..
4TTTI
period's assignment. While it may be difficult to point to instances in
which these rules are employed explicitly, they represent a natural tendency
of management to compensate employees for previous inequities.
The following table summarizes the behavior of the five rules under criteria
,. P1 - P3 (all limits are limits in weak convergence).
-4..
Ri R.2 R3 R4 F.5P. lim n 1/2 n c a C a C a C a C
o , r =2 3 4. s
P2 (T Xn > YnO} D
P3 liN/n L I L L' L Lk%',, B. 11 1 2 2 2 2
4.--.
,
In th, above table, the limit r.v.'s C. L I and L have distributions given by1 2
P{C <x} (2/w) 1/"f exp(-t2 /2)dt; x > 0
P{L< x} - 0; x < 0
1/2; 0 < x < 1
l ; x>1
P(L< x< z 2(arcaln(x1/2 )/r, 0 < x < 1.,
and
%' -9-
%.. :'..-,
%...
.4
at - (var(D,))1/2
03M(E2 1/2
a' (ED2 - (ED: E.D .)/P(D . 01)1/2i i
as a (ED" - (EID 1)2)l/2
While the rules R1. R2. R3. and R4 have some intuitive appeal as "fair"
assignment rules, we see from the table above the none of these simple rules
satisfies performance criteria necessary for a truly equitable assignment
rule. First,. all the rules have the property that the cumulative difference
In acuity measures (Q) grows as the square root of n or faster (in the case
of RI). In the case of property P2 we find that if. on a given trial, one of
the nurses is ahead of the other nurse in cumulated acuity measure, then in
the case of R1, R2 and R3 the e) ,oted time required to equalize the cumulate
acuity measures is Infinite. In the case of the third property, we wish to
Imow the fraction of the time that one nurse finds her/himself with a higher
cumulative acuity measure than the second nurse. Ideally. this random vari-
able would converge to a distribution with a single atom at 1/2. We see that
none of the rules has this property. In the case of R1, the distribution
converges to a single atom at 0 or 1; in the other Cases., the convergence is
to the *roain law. which guarantees that although the limiting distribution
Is symetric, 1/2 Is the least likely region for the limit random variable.
-10-
i ... % . " .'" .-, .. ..: . -.. - ' " ""-., . " '' " ' '' ' '" .--*,i -:
With the exception of the RI4 entry for criterion P1, all entries of the table
may be verified by routine application of the central li.it theorem (CLT) for- .,
i.i.d. r.v.'s, classical random walk results (Theorem 8.4;.4 of Chung. (197T)),
and the functional form of the arosin law (Billingsley, (1968) p. 80).
One merely uses the following representations for Zn:
n(1). Z n -1 ) E Di)
(32). Z - (2 -)( D21 - D2 )i-1
(33). Z - (21 - )
(0)." 2n z2 2 - 21A- M l 21- ID 21)
Miere A. (D I >0 , & a 1) U (D1 < 0,Qz C 0). The R4 entry uses the fact
that Zn then has the form
a Iiz-( 2 1T ) Z
where A has been defined above and I.t) mazx(k:Sk 1 i), where S1 a 1 and
S k I uf( > S k:D n> 0)
To obtain a CLT for Zn, note that
-~U-1 k k+
zW (2 1 - M)ID I* Z (-I) E D)iA k-I Jm'k+1
n-I k( 21 A - 1)(ID I + X(-I) Bk)
.-.
. " % %
where ( k:k > 1) is i.i.d. Now, 0(ZS ) - no'(a I and
a'B-'DE S + ) 12(S )2Aa a '(0 -(D I M' s - S,)(ED) + 2ED (ECS -S )B )An
where 0 B = Bk - (Sk+1 - Sk)EDI; the above equality is the second moment
version of Wald's identity. To simplify the above expression, observe that
32 - S is geometric so
*5 2 1
;(S - S) - l/P(D, > 0)
CI(S- s S P(D < O)/P(D > 01'.
Also,
(S 2 - S )0 Is, - S - k - k(E(DI[D 01 + (k-l)E(D ID < 0))
s0
1(S 2 - S )01 (ED + E(DIICD 1< 0 ))/P(D > 01'.1 1
Substituting the above relations in our expression for o (B ) and simplifying.
we find that a2 (Z S ) = nc2/P(Dk > 01. Application of the classical CLT there-
fbre proves that
a-1/2z -0 (/P(D 0)1/2 N(0,1)s -
as n0 *-, where ->denotes weak convergence, and 1(0,1) is a normal r.v. with
zero mean and unit variance. Since t(n)/n ->P(D I 0) as n . -, we may applyI-
Theorm 7.3.2 of Chung (1974) to conclude that
Lm-)1/2S( . (oIP(Dk :- 0))1 2 1(0,1)ZSL(U) '0
-12-
N~\ I., : .7 Z .- C. %* I*..!--**5 .
*,fi .O "' .''' Y , " *". * 5 " .. ".. ° .. ;'.' V . * . "~ ". " * ". , ,""" . . .x. ,* .. . *• *.." ."" ',",. : "..',S.' ,,5 -. ,'. .... .,..,. . .. . ,. .'.'.-. . .. . ..... , ..",- . . . ....-..-.-.-..-. .....-•. .. - -• .. -.,-.- . ......... "..... .-. .•. .-.. .-.
as n.-. Noting th.at ni-1/ 2 Z,Z3 .-0, a standard argum.ent thenZSI (in) a
yields
-- 1
z/2 N (0,1)
as n Which Is P1: for P3. we apply the continuous mapping theorem to an
invariance principle version of the above CLT.
The following inequalities are easily proved a 2 < ,r3 a 4 a 29 a 5< aI
Thus, in terms of criterion P1, R1. 3, and R2 are the worst, second worst,
and third worst decision rules, respectively. Somewhat surprisingly. it can
be shown by example that both a a. and 0 S < are possible - the direc-
%1
tion of the Inequality (and thus the P1 performance) depends on the joint
distribution of (V I. U1).
3. Counter-current Decision Rules: Definition and Basic Properties
Consider the deterministic situation h ere Dn '. > 0. We then have
(3.1) Lea: The rule ( :I I<) Which minimizes ( I z i is given
£ --k-1
by
*~ *1 (u(1/2)-'k ak (Zk cOtkl
Proof: Observe that
-13-
-4edtidwrtdoso uerseo/ey oehtsrrsnli e
-.mV
* 4
J..°
a a kS1. - u Z IE (20 - 1)1k-_ k-I k-i
We alaim that for each k > 1,
k k+11 2o - 1l + I E 2 , - II > IJ- -i J-1i
for it either term vanishes, then the other term must equal 1. Hence,
(3.2) zI1 . juk-i
if n a 2j or 2j - 1. Nov. it In trivially verified that the rule given In
the lmma attains the lower bound.II
The above rule behaves nicely for deterministic sequences. For exmple,IzJ remains bounded (see P1), E(TmI 5 - Yn z 1 (see P2), and N n - 1/2
kI n(see P3). This behavior suggests that one should try to generalize the
rule to the stochastic case.
(3_3) Definition: We call the rule defined by
s a "(VI < l/2)'a " 1 (Z_ < 0
the strict counter-current decision rule, and the rule
A ~ l'(Us 5 1/2)
On 0, D > 0 +la}
-14-
F or..............................................................................
the counter-current decision rule.
The strict counter-current decision rule is strictly non-anticipating and the
44cocuter-ourrent decision rule Is anticipating. We nov proceed to derive
certain asymptotic properties of (Z,:n > 1) under the two decision rules.
First, we examine the counter-current decision rule. Note that for n >2,
(3.4) z 0+ Z n- (sign Z n) ID n~1I
kernel Is given by
P ('s ~z+,!KIU )
0
*er* OWx a P(tDotl.x). It is easily verified that
h(a ZI I -Glb
is a stationary density for (Z2,:n > 1) (see Feller, (1971) p. 206) for
% -15-
- - -- - : . * i .
- - -- - , . ., - . . . , - ..-.4 .
.-4.
oalculation of a closely related density). For further analysis of this
H.C., it is convenient to introduce the following notion.
(3.6) Definition. A N.C. (Z :n > 1) on R is said to be X-irreduciblen
If there exists a probability measure 1(.) such that if A () > 0 (E a Bore1
set), then
£ 2-*P(Z nEIZ1 - Z) > 0am].
for all z.
(3.7) Lemma: Under Al - A4, then N.C. (?b:n > 1) defined by (3.4) is
I-irreducible.
Proof. Taking X to be normalized Lebesque measure on (-K/2,K/2) we
observe that by choosing n a [z/K] + 2, ([] denotes greatest integer) one
obtains)
Nz n-IzI -X} > o
fbr ny E for which ACE) > 0.11
We ean now prove the following ergodic theorem.
(3.8) Theorem. Let (Zn:n > 1) be defined by the counter-current decision
rule, and suppose that Al - A4 hold. Then, for any function k(.) satisfying
J71k(y)j-h (y)dy <
-16-
._j', ' , .', , , % %"'. ". ',,, , . .. ' .- ... .. . 1 ..- . . -. .. , ..........
I I 5ma
-: : - ' -" ".-
it follows that
'.5
(3.9) _E Mk(Z,) k(y)h (y)dy &.a.
Proof. By Lemma 3.7, we have that Z- is A-irreducible; thus h (.) is the
unique stationary probability of (Zn:n > 1) (Revuz (1975)). It follows that
5It Z has the stationary distribution, then (Z :n > 1) is a 3tationary ergodic
sequence (Ash (1972)), so one may apply Birkhoff's ergodic theorem (Lamperti
(1977). p. 92) to conclude that
I h(s)P{- Z k(Z )- k(y)h (y)dyZ - } - I
(3.10) i..e. P(I a k(Zj) f k(y)h (y)dylz - 3} - Iaujai - I
for a.e. z [-X.K] (observe that by A3, h(.) vanishes for jIz > K). But if
Z evolves according to the counter-ourrent decision rule, then IZ I I D IaI a
ad so Z Is concentrated on [-K,K and has a density there. Thus. (3.9)1
follows imediately from (3.1o).II
In particular, setting k(z) a I[to .] ), it is immediate that
%/n 1/2 a.s.
Thus, the counter-current rule leeds to a Z sequence with a *good" P3
-17-
%% V:.. :,%,.:,%' ..V .,%:.. , . .. * ... .5- .5. * ... .-. . .*. .%.."..'
.5. 4%....
-, S - - . . . . - - - . °' . 4 . • ° ' ~
property. Also, (3.5) implies that Z remains "bounded" in some sense, and
thus Improves on decision rules R1 - R5 (in which I Znj grows at rate nl/)
A for criterion P2, observe that on Za > 0),
Ta ain1k:ID+l1 + ... +lD+kl > Zd
fron which it follows that E-T nZ n) on Z. > 0) is given by M(Z) + 1.
where MWi) is the renewal function given by
MWi) - max(k:ID I + ... + ID k1 < x).
* Thus, by the elementary renewal theorem, ETnIZ) is ia symptotic to Zn/EIDaI
for ZA large. Hence, (Zn :n > 1) performs well under criterion P2 when a
oOutaer-current policy Is followed.
We turn now to the asymptotic behavior of Zn under the strict counter-
current policy. Once again, Zn:n > 1) is a Harkov chain, this time
defin*d recursively by
(3.11) Zn+1 a Zn - sIgn(Z n ) De+1
Our first order of business is to show that Zn possesses a stationary
distribution uder the strict counter-current rule. Let r. I Zn;
observe that ro satisfies, for n > 1.
(3.U) r 2+ t a D- +l
-48
I
A
We shall need the following result.
(3.13) Lema. Le t n be defined by
, [(Ca D +1+ for a > 1. with "D I. Then,un+l nl
rk. I C + K
Proof. For n a 1, the result is trivial. Proceeding by Induotion,
assume the inequality holds for n k k, and consider:r rk+ mx(rk - Dk+l, Dk+l - r
D -
4._ m rL- D k1 , K)
+ K- D+,. K)
Uma(K + (CL- DIL 1 1~ K)>1_K + [C - Dk+I+
SK + C+ 1
The next result follow easily from Lamma 3.13.
(3.14) Proposition. The N.C. (Zn:n 11) defined by (3.11) possesses a
stationary distribution.
Proof. First, observe that since MDk > o, the process Cn possesses a
limiting distribution (Kiefer and Volfoit: (1956)). As a consequence, for
€ > 0, there exists K¢ such that
laf P(%k < Kr) :0 Ck
~-19-
ir,41 . . e ~ , r e . e , . e ,W * ' o ... o * * * * * . . * -. . . . .. , . o- . . * m * * . . " . . q ]
.. , ,. , ,, , ' L.. . I ,-4. . S ... .. . . . ,. . . - .o - . . • .
from which it follows that
n- . r { k I (K + K > I -ck
Thus, the probabilities Pl'k-:,) are tight (see Billingsley (1968), p. 37),
" from which it follows that the probabilities P(Zk -) are "ight. A glance at
(3.11) shows that the recursion is continuous in Zn, and hence a well-known
theorem on weakly continuous kernels may be applied (see, for example, Karr,
(1975)) to conclude that ZM possesses an invariant probability.
Given Lemma 3.13 and Proposition 3.14, the proof of the following theorem
follows the same pattern as that of Theorem 3.8.
(3.15) Theorem. Let (Zn:n > 1) be defined by the strict counter-current
decision rule, and suppose that Al - A4 hold. Then, Z possesses a uniquenU
stationary density h and for any k(.) satisfyinga
J71 k(y) I h (y)dy <
It fbllows that
(316 01 k(Z )-f k(y)h (y)dy as
Observe that if ha(z) is stationary for Zn . then so is h (-z). Thus, by
uniqueness of the stationary distribution, h Iz) h (-z) and therefore
ba (S)dz - 1/2.
So we have, by applying (3.16), that N U/n-1/2 a.s.
-20-
' .%
%. 44 ,- ... :'i -,,;.,.,:''' .v.-..'S'.-."'''. ...- : " i.'- . ,i .'""- ". .".- ,':.:."2g '"." " ... ':'2rj'."'""'
ad therefore the strict counter-current rule behaves well in terms of both
criterion P1 and P3 ((3.16) says that IZ.1 remains "bounded").
As in the case of the counter-current rule, the analysis of property P2
requires the representation
T -in(k:D +l + +D+ k > Z }
which is valid on (Z n> 01. Although the D Is are not positive r.v.s, it
Is still true that the renewal-type result
£(TUjIZ a)%InED
bolds for Z large. If U(mm) z D U+1 D " I Wald's equality Impliesthat E(U=n.T n ) • E(T n • )t the boundednes of the Dk's implies
that zn _ U(n.To ) C + K. from which the asymptotic relation follows.
Thus, the strict counter-current policy behaves well under P2.
Finally, we shall show that introduction of "noise" into the problem always
leads to a degradation in the behavior of tounter-current-type rules. From
(3.5) w have that
"y lr h ( y ) d y . E D 1r + "
-m (r+l)EID0 I
fbr r > 0. In particular, taking r 1 and applying (3.9), we obtain
)- ED' ED var(D )
nk-i 2EIj 2 EIDI
M.S. From Lea 3.1, it follows that EDn/2 is the deterministic lower bound;
.1 ~-21-
V
, , , ' , , L • .. . . . . • . . . .. K.:.. • . • • . . .. . .. .. . .,:-:, .. ..
introduction of stochastic noise leads to the presence of an additional
positive term given by var(D )/ED
This discussion also carries over to the strict counter-current decision rule.
The argument in this case centers around (3.12). If r has the distribution
of IW, where Z has stationary distribution h , then from (3.12),
(3.18) r a Ir - DI
(Qdenote3 equality in distribution), where F and D are independent. It is
evident from Lemma 3.13 that
'(r > x) < P(f.+K> x)
where & is the limiting distribution of Ck" It is well-known that under A3,
BEk < for all k. (Kiefer and Wolfowitz (1956)). and thus Erk < -for all k.
So, we square both sides of (3.17) and cancel common terms to obtain
X - I ylh (y)dy -122 2EDn+I
Thus, we obtain that under the strict counter-current decision rule,
., ED n1a~s(3.19) a E I ZkI4 -~ a--1 " 2 g n+1
It Is worth observing that since ED < EIDI . the strict counter-current rulen n1
behaves worse than the counter-current rule under the "sum of the absolute
value" criterion (compare (3.17) and (3.19)).
-22-
% % % , -k
.4':
4. Optimality of Counter-current Decision Rules
In Section 3, we saw that counter-current decision rules enjoy a number of
desirable properties - in this section, we shall show that counter-current
rules possess certain optimality characteristics.
- (4.1) Theorem. Let (Z :n > 1) be constructed according to the counter-
current decision rule. Then, under Al - A4, the counter-current decision
rule minimizes a.s. both
n+T -1n
1.) E q(Zk ) on (Zn > 0), where q(.) is any increasing functionk-n
ii.) -T-IZl I l" i'- =1 Z. -n
over the class of no-anticipating decision rules.
Proof. For i.), suppose that the counter-current rule is first violated
St time k, where n <k < n + Ta. Then, for k< m < n +T . we have
(4.2) Zu _ Zkl1 + IDI - IDk+l .....- ID aI - Z, + IDk
'4..,
utere Z' is constructed from the counter-current rule. Inequality (4.2)
imediately implies (4.1) i.).
For 11.), it is clearly sufficient to prove that for any non-antioipating
decision rule,
(4.3)-imlZnl > K a...;
-23-
S.,2
-," .. ''- ' ,-', -, -' ,' ,. ." * . • -' - r ," *.* * • ,- - . .. . . - .. -. . * ,. . . ~~
----------------------....... : ' '
the result then follows since the counter-current decision rule clearly
attains the lower bound of (4.3).
Note that for fixed c > 0, and any non-anticipating decision rule,
(4.4) P(ID +1 - KI < , ID+ 2 - K - %1 < C,
ID n+3 - fj< cjIZIO z U, *z
AA 2 A A-(F( + C) - F(K -c))2 (F(Z +Kn + C) - F(Zn + K-c))
> C min iCF(J/2) - F(c(j - 1)/2)) 3 > 01< j </
where K = [K/eJ ([M r greatest integer function): Because of the uniformity
of (4.4) over Zn, one can apply the conditional Borel-Cantelli Lemma (Doob
(1953), p. 323) to infer that
(4.5) IoD-+l - K < , ID a K - Z I < e,n+1 ~ n+2 f
1D5+3 - [I < c infinitely often) - 1
Letting T be a generic time at which the inequalities (4.5) are satisfied,
one can easily cheek (just go through the eight different Cases for
(aT+, aT+20 fLT+ 3))such that IZT+3l I K - 3c. if IZI <_ K/2.
A similar argument to that used above shows that for any decision rule,
-24-
% V v. , ,5 ,. %5 . , , .. %. . . ,. . ..
J 1
I ZI < Ki2 must occur infinitely often, from which it follows thatn
-" IzI > K - 3C. proving ii.).11
Recall that in Section 3, we proved that counter-current policies suffer a
degradation in performance when stochastic "noise" is introduced. We shall
now show that counter-current policies are optimal for the L0 criterion,
"pthus proving that the optimal polllaes suffer in the presence of "noise."
To accomplish this goal, we will invoke the theory of Markov decision chains
on 3.
We will first deal with the strictly non-anticipating situation. The
decision chain involves two actions (1 = 0 or a z 1), and consequently two
transition kernels. A cost equal to the absolute value of the state occupied
Is charged for each transition; costs are independent of action. With this
" framework, the optimality equation for the average cost decision process is
given by
(4.6) y + ,(z) - I-I + uin(Em(z + D ,-(- - D)I
%here a(z) is the optimal return function.
(4.7) Proposition. Under Al - A4I, a solution pair (Y, m) exists to
(4.6) and is given by
- ,,*(z) - zZ/2ED
. - D 2ED
%-25-
4.
:,r,.%,, , 5,.,+'..:,,-%,.,z.,.'.p,...-.-,., ..,.... .... ,..............-.... +......... ......-.... ....... ...... .......
Proof. Observe that
min(m-(z + 0) Em(z - D ) - (z' + ED2)/2ED + :n{-zz)I I I
- ,€,) + Y - Izl.II
To give a concise proof of the next theorem, we shall consider a restricted
mclass of decision rules.
< a - 1 , , ...)lai 1Z/- - 01
Note that if a decision rule is not in A. then EZ is of order n. whichn
SIndicates that IZI is growing unboundedly in expectation; clearly this is
undesirable from a practical viewpoint. We henceforth restrict ourselves
to decision rules in A.
(4.8) Theorem. Under Al - A4.
(.)ED'
(49 7 n .1 ZIZk - 21D 1for any randomized strictly non-anticipating decision rule in A; the
minimum In (4.9) is attained by the strict counter-current decision rule.
Proof. We apply a theorem due to Ross (19681. Our class of randomized
strictly non-anticipating decision rules corresponds to the set of policies
enunciated there. Following Ross's proof, we see that for any randomized
strictly non-anticipatng decision rule (whether In or not inA), Z satisfies
-26-
N. N• :7 ,.', .:.' .' , ", ' . -' ., "' " . , """: ", , ,"" -'. , " , " - ,, T ' . 'o , '- ' .. " """" , ."" " , ."," .,%.- , ' ; , , -: ' , . ' ' . , L - , . ' ' ' . '.&_ - : -V, %" . " , - " , " % . - " , ' - . . . ' - - , . - , . . . . , '
Ee(10 -C EZ - EZ2.+ E KIZkI
2E n kol.4.o.'-
thus, If the rule Is In A. one obtains (14.9) (since EZ < Kt). As usual,
equality in (4.10) occurs if one uses the policy whioh consistently minimizes
the right-hand side of the optimality equation (4.6); this minimizing policy
is easily seen to be the strict counter-current rule. However, because of our
A restriction, we still need to show that the strict counter-current rule Is
to A
From Lamma 3.13, it follows that
C (1.11) K7t I 2(K" . EC")..4
Nov, it is ell-known that C is stoohastically Increasing to Its steady-state
C and that I C < - since ED3 < - (se Kiefer and Wolfbwitz (1956)).
.4 lMenoe, EZ2/n * 0 for the strict counter-ourrent rule.II
Ve conclude this section with a statement and short proof of the corresponding
result for the counter-current decision rule.
(4.12) Theorem. Under Al - A4.
(4.13) lI-t z l _
k-a 2310 1
for any randomized non-anticipating decision rule in A; the minimum in (4.13)
-27-
'.,4 ,'% . .'? ,., l...-...",.-.-'- ,",, .. -". . ,."-". - . . . , . . . ''"--.-. -. ... ',..v. " ' ' . .
iil• B m m Ii u lll nl~wldnnui ll ml Eia ~ n 6
is attained by the counter-current decision rule.
The key Idea here Is to define an appropriate state space for the decision
chain. We choose to use atates of the form (z~d). where the z component cor-
responds to the current value of the sum or absolute Values Of Zi's; d cor-
responds to the current value of the r.v. Dkc. Letting m(z.d) be the optimal
* return function for the decision chain, it is easily seen that the optimality
equation nov takes on the form
Y + m(s~d) - minfix + dl + EmC: + d, D ),is - dl + Em(z - d, D).
The pair (Y,m) which solves the above equation is given by
y a D02 /291D I
a(z~d) - (lxi - ldl)1/2E1D1.
again. the counter-current rule is the minimizing rule for the optimality
equation. Since Iz:lj < K under the counter-current rule, It lies in A&.
'I" Note that the optimality results given above show that the optimality of
counter-current decision rules does not depend on the detailed form of the
distribution of the Dk's. Hence, one expects these results to be quite
robust In practice.
SUMMNARY AND CONCLUSIONS
In the previous sections we have formalized an important class of management
decision problems, investigated the properties of a nuber of intuitively
-28-
e r.. or* 0 %% 4,.
. .
appealing assigrment rules and developed a new class Ot assignment rules. the
counter-ourrent policies. We have argued. by citing a number of examples,
that the decision problem under investigation finds application In a wide
variety of circumstances.
We have shown that ordinary intuition is often o little assistance in
developing equitable assignment policies. It is remarkable that none of the
easily conceived assignment rules will assure reasonable local or asymptotic
properties. (This raises a number of interesting psychological and philoso-
phical questions about our ability to make Informed judgments about the pro-
perties of stochastic processes from the properties of the "generating"
mechanism.) We have shown that among the rules considered, rule R1 is worst,
followed by R3 and R2. Rules I and R5 are better than the others, but one
cannot kmv which of these is "better" without exaining the distribution of
the D Is In some detail. "Better" refers to the rate of growth of IZ 1. with
faster growth being "worse." To restate the results In less formal terms, we
• se that "Fixed Assignment* Is worse than "Random Assignment," which in turn
Is worse than "Fixed Alternation" (or "Block Rotation"). Both o the rules
itloah attempt to reverse the last period's inequities (i.e. R4 and R) are
better than the other rules. An Interesting new result emerges: under cer-
tain circumstances the "clairvoyant" rule, R5, is interior to R4, a rule that
does not assume knowledge of the severity o the assignments in advance.
Despite the disappointing performance ot the more obvious rules we have
proposed a class of rules (counter-current), Which not only have the desired
asymptotic and local properties but also are optimal in the sense described in
-29-
a. ~~V N Va~
Sections 3 and 4.
A Important question arises at this point. Is it possible to extend these
results to the n-person assignment problem? The negative results of Section
2 almost certainly will arry over Into the multi-person problem. More
Importantly. Is there a multi-person analogy far the counter-current rule?
Does It have the same properties as the two-person cane? In the n-person
analogue to the counter-current rule. one assigns duties in any period in the
.4 following way: the Individual with the largest cuulative workload measure
to date receives the assignent with the smallest value or smallest expected
value. That Individual is removed from the list of individuals to be con-
sidered and the rule is repeated with n-1 remaining assiginents. It can be
shown relatively easily from the results of Section 4 that this n-person
generalization of the couter-current rule has at least the following
desirable property.
Let c?..., ) be the duty assignments on units 1. ... , n In period t
and we &asme that the joint density of (It, .. ),io everywhere positive
amnII K end il~ < K for every I a 1,... n, and every t a 1, 2.
If the I-vectors are i .1.d., then for any assignment rule
there Utis the sun of all workload measures for Individual I through time
t. Fuarther, the lower bound of this Inequality Is obtained by using the
a-person counter-current rule discuased above.* This follow from the fact
that if Uj i4 then the assignment of Individual .1 will be greater than
-30-
~ ~~ ~ %, 1 .. , .,
.. J
that to individual I at time t . 1.
A number of related mathematical and psychological issues deserve additional
Investigation. The n-person case presents interesting challenges. Consider
areas of application such as equitable distribution of merit pay among univer-
sity faculty or public school teachers. These issues require more complex
models since they presuppose inherent inequalities in "true" performance.
Vthough they do share characteristics with examples we have discussed.
ACKNOLEDENT
'p The research of the first author was sponsored by the United States Army
under Contract No. DAAG29-80-C-O041. The authors gratefully acknowledge the
contributions of Professor Dennis Frybeck and Dr. Edwardo Lopez to the work
that led to this paper.
Adama, J.S. 1963. "Toward am Understanding of Inequity," Journal of
Abnormal and Social Psychology, Vol. 67, pp. 422-36.
1965, "InJustice in Social Exchange," in Advances in
Experimental Social Psohology. Academic Press, New York, pp. 267-299.
Andreva, I.U. and Henry, N.M. 1963, "Management Attitudes Toward Pay.
Industrial Relations, Vol. 3, pp. 29-39.
-31-
% '6L';,-,'" "", _. :.'",. . ,:....' -'.-:' .. '' .':-4,''. . -'.:.".' " .""" . :- ." .". " -".".. . S .-. '" S.""-."- -S
Ash, 3.B. 1972. Real Analysis and Probability, Academic Press, New York.
Billingsley, P. 1968, Convergence of Probability Measures. J. Wiley
and Sons, New York.
Chung, K.L. 1974, A Course in Probability Theory, Academic Press, New
York.
Doob, J.L. 1953. Stochastic Processes, John Wiley and Sons. New York,
p.323.
Feller, W. 1971, An Introduction to Probability Theory and Its
Applications, 2, J. Wiley and Sons, New York.
Karr, A.F. 1975, "Weak Convergence of a Sequence of Markov Chains."
Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33. 41-48.
Kiefer, J. and Wlfowitz, J. 1956. "On The Characteristics of the General
Queueing Process, with Applications to Random Walk," Ann. Math. Stat.,
27, 147-161.
Lavler, E.g. 1971, Pay and Organizational Effectiveness: A Psychological
View, icGraw-Hll, Now York.
Limperti, J. 1977, Stochastic Processes. Springer-Verlag, New York.
-32-
% %
Ii -
77. 1 -7 T-7 77.7A .3
e
.'.
Revux, D. 1975, "arkov Chains, North-Holland, Amsterdam.
Was. 3. 1968, Arbitrary State Markovian Decision Processes, Ann. Math.
t-at. .. 2118-2122.
Shapiro, N.J. and Wahba, N.A. 1978, "Pay Satisfaction: Empirical Test of the
Diarepancy 1odel," anagement Science, Volume 24, No. 6.
,,
'." '..
-33-
%* % '....
.~~~ % %. %, '.",':. ., ':,.' ...... ',....... . '-" . , , . . .,,- . '.. -... ' - .. ,.,. . , -- ,,-, ,: : . . ,.-.-. -,,.
-. -C N.7
SECURITY CLASSIFICATION OF THIS PAGE (Whm Date Bnte4
REPORT DOCUMENTATION PAGE% BEFPORE COMPLETING FORM
4. TITLE (and Subti) S. TYPE OF REPORT & PEROD COVEREDSEquitable Assigment Rules Summary Report - no specific
Slreporting period
S. PERFORMING ORG. REPORT NUMBER
7. AUTHOR(@) S. CONTRACT OR GRANT NUMSER(s)
P. Glynn and J. L. Sanders DAAG29-80-C-0041
9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT. TASK
Mathematics Research Center, University of AREA & WORK UNIT NUMBERS
: 610 Walnut Street Wisconsin Work Unit Number - 5Optimization and Large Scale
Madison, Wisconsin 53706 SystemsII. CONTROLLING OFFICE NAME AND ADORESS 12. REPORT DATE
U. S. Army Research Office May 1984
P.O. Box 12211 1s. NUMBER OF PAGES
Research Trianqle Park, North Carolina 27709 3314. MONITORING AGENCY MAME 4 AODRESS(lf diloent Ios Controllg Otfico) IS. SECURITY CLASS. (of t111 report)
UNCLASSIFIEDI5.S DECLASSI FICATION/ DOWNGRADING
SCHEDULE
I. DISTRIBUTION STATEMENT (of1 fIe Repat)
-. Approved for public release; distribution unlimited.
-: %4 17. DISTRIBUTION STATEMENT (f Me btract emtmred in Block 20, It d1ent froa Report)
IW. SUPPLEMENTARY NOTES
SO. KEY WORDS (Caulk... an reverse side if necesaranif idbnttify ' block aumbor)
Assignment rules, dynamic programming
20. ABSTRACT (Cmnmie an tevers ald. It necesary and Identif by bock nmber)
* This paper investigates the formalization of an important class ofX management decision problems. The problems considered are those of making
equitable workload assignments to personnel. The paper proposes a series ofintuitively appealing assignment rules, including random assignment, fixedassignment, block rotation and rules that reverse inequities caused by thelast period's assignments. It is shown that in the two-person case none ofthese rules satisfies the simple criterion that cumulative differences ofworkload assignments among personnel become and remain small. Differences in
:. FORM
DD AD 1473 EDITION OF I NOV65 IS OBSOLETE UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE (When Date Eafe.e
, . ... . .
?-777.
ABSTRACT (Continued)
the properties of these rules are investigated under three additional but les .strenuous criteria. It is shown that a new assignment rule called the
-[. "counter-current" rule does satisfy the criterion stated above; furthel,shown that it is an optimal rule under a fairly weak set of requirements. Theextension of the results from the two-person case to the n-person case tcn-person case is discussed briefly and some initial results are presented.A
.g
',p
%-6
%. .
T% o4 'o
,,,. --- , -,* - ------...... ....... ... .L"".' -" ".'....
l~.i 0 ,
'I *0A 4 ., C
4
A-
-, *~ V4
I . ~ ~
a' V4,., ''i'- K 7
~**-~ ~- ~'
' -4 :0 -
'.4- ..
1 '4-,.
~0 . *0
p .~ ~
* .K~0~
~A ~ -
,j44Z~.
oar .~.
'~i0t ' -\F .
~ j . 40~~ I N
'a'. IjrII.. $,i Lit. L-.~.
i-gas., '. g ., a* ;~;~~t .~ 34'0, a,
' *<~ '4~ ~
* V~'. '
* *0 ~
,W~* 'a~* :' -, I ~* 0 a .oa~
~ .
0~~ a'.:.
~ .~ 0~'0~ ~
* i
0%~ ~ i?',~. x
U .4-*.1
dtd
it's
- ~ 'aI 0 ,~ ,~, , .,,
.1~~ ~
~* 44v~ A.~.
4
a' *
ITT
I
k